Finite Group Databasebeta

← #34185096

Subgroups — #34185096

Order 20224.

428 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23772051C211yesmaximal, characteristic, complemented
2#23762011C211yesmaximal, characteristic, complemented
3#23761834C211yesmaximal, characteristic, complemented
4#23761817C211yesmaximal, characteristic, complemented
5#23761298C211yesmaximal, characteristic, complemented
6#23761271C211yesmaximal, characteristic, complemented
7C79×#3115C211yesmaximal, characteristic
8#19098624—22no—
9#19098620—22no—
10#19098620C2211yescharacteristic
11#19098616C2211yescharacteristic
12#19098590—21no—
13#19098530—21no—
14C4×#11929608C2211yescharacteristic
15C2×#14579072C2211yescharacteristic
16#19098303—22no—
17Q8⋊(C79⋊C8)—21no—
18D4⋊(C79⋊C8)—22no—
19D4⋊(C79⋊C8)—21no—
20#532×C79C2211yescharacteristic
21C4×#41×C79C2211yescharacteristic
22#482×C79—21no—
23#475×C79C411yescharacteristic
24#474×C79C411yescharacteristic
25#460×C79C411yescharacteristic
26#459×C79C411yescharacteristic
27#449×C79—21no—
28#442×C79—21no—
29#440×C79—21no—
30#432×C79C2211yescharacteristic
31C2×#11929608—21no—
32C22×C79⋊C8—21no—
33C4⋊(C79⋊C8)—24no—
34C4⋊(C79⋊C8)—21no—
35C4⋊(C79⋊C8)—21no—
36C22⋊(C79⋊C8)—24no—
37C22⋊(C79⋊C8)—21no—
38C22⋊(C79⋊C8)—21no—
39#14579072—21no—
40#14579072—21no—
41#14579072—21no—
42#14579072D412yes—
43C4×C79⋊C8—21no—
44C4×C79⋊C8D412yes—
45C2×#41×C79—21no—
46C2×#41×C79D411yescharacteristic
47C2×SD16×C79—21no—
48C2×D8×C79—21no—
49C4⋊Q8×C79C2×C411yescharacteristic
50#127×C79C2×C411yescharacteristic
51#124×C79—21no—
52#121×C79—21no—
53#121×C79—21no—
54C4×Q8×C79C2×C411yescharacteristic
55C4×Q8×C79D411yescharacteristic
56C4×D4×C79—21no—
57C4×D4×C79—21no—
58C4×D4×C79—21no—
59C4×D4×C79—21no—
60C4×D4×C79C2×C411yescharacteristic
61C4×D4×C79D411yescharacteristic
62#117×C79—21no—
63#117×C79D411yescharacteristic
64C2×C42×C79—21no—
65C2×C42×C79C2311yescharacteristic
66#107×C79—21no—
67#106×C79—21no—
68C4⋊C8×C79C2×C411yescharacteristic
69C4⋊C8×C79C2×C411yescharacteristic
70#103×C79—21no—
71#102×C79—21no—
72#102×C79—21no—
73#102×C79—21no—
74C22⋊C8×C79—21no—
75#97×C79—21no—
76#11929608—41no—
77#11929608—41no—
78C2×C79⋊C8—22no—
79C2×C79⋊C8—22no—
80C2×C79⋊C8—41no—
81C2×C79⋊C8—41no—
82C2×C79⋊C8—41no—
83C2×C79⋊C8—41no—
84#41×C79—41no—
85#41×C79—41no—
86#41×C79—82no—
87C2×Q8×C79—21no—
88C2×Q8×C79C22⋊C411yescharacteristic
89C2×D4×C79—21no—
90C2×D4×C79—21no—
91C2×D4×C79—21no—
92C2×D4×C79—21no—
93C2×D4×C79—41no—
94C2×D4×C79C22⋊C411yescharacteristic
95C22×C4×C79—21no—
96C22×C4×C79—21no—
97C22×C4×C79—21no—
98C22×C4×C79—41no—
99C22×C4×C79C2×D411yescharacteristic
100C22×C4×C79#4111yescharacteristic
101C22×C4×C79#4111yescharacteristic
102SD16×C79—42no—
103D8×C79—42no—
104C2×C8×C79—21no—
105C2×C8×C79—21no—
106C4⋊C4×C79—21no—
107C4⋊C4×C79—21no—
108C4⋊C4×C79—21no—
109C4⋊C4×C79—21no—
110C4⋊C4×C79C22⋊C411yescharacteristic
111C4⋊C4×C79C22⋊C411yescharacteristic
112C22⋊C4×C79—21no—
113C22⋊C4×C79—21no—
114C22⋊C4×C79—21no—
115C22⋊C4×C79—41no—
116C22⋊C4×C79—41no—
117C42×C79—21no—
118C42×C79—21no—
119C42×C79—21no—
120C42×C79C22×C411yescharacteristic
121C42×C79C2×D411yescharacteristic
122C42×C79C2×D411yescharacteristic
123C42×C79C2×D411yescharacteristic
124C23×C79—21no—
125C23×C79—41no—
126C23×C79#12311yescharacteristic
127Q8×C79—21no—
128Q8×C79—21no—
129Q8×C79—81no—
130D4×C79—21no—
131D4×C79—21no—
132D4×C79—42no—
133D4×C79—42no—
134D4×C79—41no—
135D4×C79—41no—
136D4×C79—81no—
137D4×C79—82no—
138C2×C4×C79—21no—
139C2×C4×C79—21no—
140C2×C4×C79—21no—
141C2×C4×C79—21no—
142C2×C4×C79—21no—
143C2×C4×C79—41no—
144C2×C4×C79—41no—
145C2×C4×C79—42no—
146C2×C4×C79—41no—
147C2×C4×C79—81no—
148C2×C4×C79—81no—
149C2×C4×C79—82no—
150C2×C4×C79C2×C22⋊C411yescharacteristic
151C2×C4×C79C4×D411yescharacteristic
152C2×C4×C79C4×D411yescharacteristic
153C2×C4×C79#12011yescharacteristic
154C2×C4×C79#12111yescharacteristic
155C2×C4×C79#12111yescharacteristic
156C8×C79—41no—
157C8×C79—41no—
158C79⋊C8—42no—
159C79⋊C8—42no—
160C79⋊C8—81no—
161C79⋊C8—81no—
162C22×C79—21no—
163C22×C79—21no—
164C22×C79—21no—
165C22×C79—41no—
166C22×C79—41no—
167C22×C79—41no—
168C22×C79—81no—
169C22×C79—81no—
170C22×C79—81no—
171C22×C79#38611yescharacteristic
172C4×C79—21no—
173C4×C79—21no—
174C4×C79—21no—
175C4×C79—21no—
176C4×C79—21no—
177C4×C79—41no—
178C4×C79—41no—
179C4×C79—41no—
180C4×C79—81no—
181C4×C79#42111yescharacteristic
182C4×C79#42111yescharacteristic
183#9313—791nomaximal, Sylow
184C2×C79—41no—
185C2×C79—81no—
186C2×C79—161no—
187C2×C79#188111yescharacteristic
188C2×C79#189711yescharacteristic
189C2×C79#190311yescharacteristic
190#3115D7911yescharacteristic
191#2923—791no—
192#1530—791no—
193#1488—791no—
194#1484—791no—
195#1302—791no—
196#1291—791no—
197C79#931311yescharacteristic, complemented, Sylow
198#532C2×D7911yescharacteristic
199C4×#41C2×D7911yescharacteristic
200#482—21no—
201#475Dic7911yescharacteristic
202#474Dic7911yescharacteristic
203#460Dic7911yescharacteristic
204#459Dic7911yescharacteristic
205#449—21no—
206#442—21no—
207#440—21no—
208#433—1582no—
209#432—791no—
210#432—1582no—
211#432C2×D7911yescharacteristic
212#431—791no—
213#424—1581no—
214#408—1581no—
215C4×M16—791no—
216C2×#97—791no—
217#329—1582no—
218Q8⋊C8—1581no—
219D4⋊C8—1582no—
220D4⋊C8—1581no—
221C2×#41—21no—
222C2×#41C22⋊D7911yescharacteristic
223C2×SD16—21no—
224C2×D8—21no—
225C2×M16—1581no—
226C22×C8—1581no—
227C4⋊Q8C2×Dic7911yescharacteristic
228#127C2×Dic7911yescharacteristic
229#124—21no—
230#121—21no—
231#121—21no—
232C4×Q8C2×Dic7911yescharacteristic
233C4×Q8C22⋊D7911yescharacteristic
234C4×D4—21no—
235C4×D4—21no—
236C4×D4—21no—
237C4×D4—21no—
238C4×D4C2×Dic7911yescharacteristic
239C4×D4C22⋊D7911yescharacteristic
240#117—21no—
241#117C22⋊D7911yescharacteristic
242C2×C42—21no—
243C2×C42C22×D7911yescharacteristic
244#107—21no—
245#106—21no—
246C4⋊C8—1584no—
247C4⋊C8—1581no—
248C4⋊C8—1581no—
249C4⋊C8C2×Dic7911yescharacteristic
250C4⋊C8C2×Dic7911yescharacteristic
251#103—21no—
252#102—21no—
253#102—21no—
254#102—21no—
255C22⋊C8—21no—
256C22⋊C8—1581no—
257C22⋊C8—1581no—
258C22⋊C8—1584no—
259#97—21no—
260#97—792no—
261#97—1581no—
262#97—1581no—
263#97—1581no—
264C4×C8—792no—
265C4×C8—1581no—
266#41—41no—
267#41—41no—
268#41—82no—
269C2×Q8—21no—
270C2×Q8C22⋊Dic7911yescharacteristic
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—41no—
276C2×D4C22⋊Dic7911yescharacteristic
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—41no—
281C22×C4C2×C22⋊D7911yescharacteristic
282C22×C4#1192962311yescharacteristic
283C22×C4#1192962311yescharacteristic
284SD16—42no—
285D8—42no—
286M16—3161no—
287M16—3161no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—3161no—
291C2×C8—3161no—
292C2×C8—3161no—
293C2×C8—3161no—
294C2×C8—1582no—
295C2×C8—1582no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4C22⋊Dic7911yescharacteristic
301C4⋊C4C22⋊Dic7911yescharacteristic
302C22⋊C4—21no—
303C22⋊C4—21no—
304C22⋊C4—21no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C42—21no—
308C42—21no—
309C42—21no—
310C42C22×Dic7911yescharacteristic
311C42D4×D7911yescharacteristic
312C42D4×D7911yescharacteristic
313C42C2×C22⋊D7911yescharacteristic
314C23—21no—
315C23—41no—
316C23#1457914211yescharacteristic
317Q8—21no—
318Q8—21no—
319Q8—81no—
320D4—21no—
321D4—21no—
322D4—42no—
323D4—42no—
324D4—41no—
325D4—41no—
326D4—81no—
327D4—82no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—82no—
340C2×C4C2×C22⋊Dic7911yescharacteristic
341C2×C4D4×Dic7911yescharacteristic
342C2×C4D4×Dic7911yescharacteristic
343C2×C4#1457912911yescharacteristic
344C2×C4#1457913311yescharacteristic
345C2×C4#1457913311yescharacteristic
346C8—41no—
347C8—41no—
348C8—3162no—
349C8—3162no—
350C8—6321no—
351C8—6321no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22#1909844311yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—81no—
371C4#1909858011yescharacteristic
372C4#1909858011yescharacteristic
373C2—41no—
374C2—81no—
375C2—161no—
376C2#2376376411yescharacteristic
377C2#2376384911yescharacteristic
378C2#2376388211yescharacteristic
subgroupG/H
centerC22#19098443
commutatorC42×C79C22×C4
FrattiniC2×C42C22×D79
Fitting, prime radicalC79×#3115C2
socleC22×C79#386
cosocleC2×C42×C79C23
hypercenter#3115D79
hypocenterC79#9313
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3115

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #23772051C2semidirect
2. #23762011C2semidirect
3. #23761834C2semidirect
4. #23761817C2semidirect
5. #23761298C2semidirect
6. #23761271C2semidirect
197. C79#9313semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#237720512
#237620112
#237618342
#237618172
#237612982
#237612712
C79×#31152
#931379